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axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.
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Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.
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Kelly, G.M (1964). "On MacLane's conditions for coherence of natural associativities, commutativities, etc".
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for the case of an abstract category, since they follow directly from the axioms; in fact, they
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Collection of conditions requiring that various compositions of elementary morphisms are equal
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360:{\displaystyle \alpha _{A,B,C}\colon (A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)}
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Actually, there are many ways to construct such a morphism as a composition of various
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is a collection of conditions requiring that various compositions of elementary
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are equal. Typically the elementary morphisms are part of the data of the
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In these two particular examples, the coherence statements are
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Laplaza, Miguel L. (1972). "Coherence for distributivity".
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1276:We have now the following coherence statement:
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968:are equal.
204:mathematics
154:introducing
88:. See the
1430:1911/62865
1334:References
275:associator
50:improve it
1263:) :
938:α
911:⋯
895:⊗
879:⊗
876:⋯
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414:α
349:⊗
340:⊗
334:→
328:⊗
319:⊗
310::
289:α
243:α
216:morphisms
90:talk page
56:talk page
1499:Category
1440:(1971).
1323:theorems
1234: :
1179: :
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980:Identity
220:category
372:objects
150:improve
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1243:, and
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135:, or
1460:ISBN
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1200:and
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210:, a
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